Strichartz estimates for Schr\"odinger equations with variable coefficients and unbounded potentials

TL;DR

This paper establishes sharp local and global Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials, extending previous results to more general long-range and growth conditions.

Contribution

It provides new sharp local-in-time and global-in-space Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials, including endpoint cases and derivative loss considerations.

Findings

Sharp local-in-time Strichartz estimates outside large compact sets.

Global-in-space estimates under nontrapping conditions.

Strichartz estimates with small derivative loss without asymptotic flatness.

Abstract

The present paper is concerned with Schr\"odinger equations with variable coefficients and unbounded electromagnetic potentials, where the kinetic energy part is a long-range perturbation of the flat Laplacian and the electric (resp. magnetic) potential can grow subquadratically (resp. sublinearly) at spatial infinity. We prove sharp (local-in-time) Strichartz estimates, outside a large compact ball centered at origin, for any admissible pair including the endpoint. Under the nontrapping condition on the Hamilton flow generated by the kinetic energy, global-in-space estimates are also studied. Finally, under the nontrapping condition, we prove Strichartz estimates with an arbitrarily small derivative loss without asymptotic flatness on the coefficients.